Imagine that you are an engineer designing a new electrode tip that
will be used to monitor brain activity in people with epilepsy. The
end product will be an array of electrodes that will be implanted in
the area of a patient's brain where doctors believe the epileptic
seizures begin. Once the array of electrodes is implanted in the
brain, doctors will be able to monitor the brain's activity before and
during a seizure to pinpoint exactly where the seizure begins.

Currently, for severe cases of epilepsy that don't respond to other
treatment, the recommended treatment is removing the small part of
brain tissue that starts the seizure. Since doctors want to remove as
little of the brain as possible, implanting electrode arrays is
necessary to find exactly what part of the brain is causing the
trouble.

You, however, are working with a team of scientists looking for
methods to stop the seizure that don't involve removing parts of the
brain. The idea is the following. The electrode arrays can detect
patterns of brain activity that signal the seizure is about to begin.
What if one can use the electrodes to disrupt that pattern of activity
so it doesn't develop into a full blown epileptic seizure? If one
passes small pulses of current through the electrodes with the right
timing, the hope is that a seizure can be avoided.

With this goal and many other technical requirements in mind, you have
developed a prototype electrode tip whose shape is given by the solid
$\dlv$ diagrammed below. You have optimized the electrical properties of
the electrode tip both through developing the shape $\dlv$ and through
incorporating different metal alloys along different parts of the
electrode.

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A three dimensional domain of an electode trip. The solid $\dlv$ represents a tip of an electrode.

You are quite excited about your new design. You show it to the chief
scientist on the project. She likes your design, but has one
concern. She wants you to calculate the total charge that may build
up on the electrode to make sure it doesn't reach unsafe levels.

From the metal composition of your electrode tip and the information
the chief scientist gives you about the current, you quickly
calculate $g(x,y,z)$, the charge density at each point $(x,y,z)$
within the electrode tip. To find the total charge, all you need to
do is calculate the triple integral
\begin{align*}
\text{total charge } = \iiint_\dlv g(x,y,z) \, dV,
\end{align*}
where $\dlv$ is the solid representing the electrode tip.

You struggle with calculating the integral, but, especially given the
complicated shape of the electrode, you find that the integral is too
difficult to solve directly. Then, you remember something you learned
in your multivariable calculus course: changing variables in triple
integrals.

Step 1: Integrate over new region $\dlv^*$

Instead of trying to directly integrate $g(x,y,z)$ over $\dlv$, you
realize you could solve your problem by finding a change of variables
\begin{align*}
(x,y,z) = \cvarf(\cvarfv,\cvarsv,\cvartv)
\end{align*}
that maps a simpler solid $\dlv^*$ onto the complicated solid $\dlv$. Then,
rather than integrating over $\dlv$, you could integrate over $\dlv^*$.

After a long night's worth of calculations, you come up with a
function $\cvarf(\cvarfv,\cvarsv,\cvartv)$ that looks promising. The effect of the
function $\cvarf$ is demonstrated below. The function $\cvarf$ maps a
simple solid $\dlv^*$ (shown in the first panel, below) in $(\cvarfv,\cvarsv,\cvartv)$-space to the electrode
tip $\dlv$ (shown in the second panel, below) in $(x,y,z)$-space. We often say that
$\cvarf(\cvarfv,\cvarsv,\cvartv)$ parametrizes $\dlv$ for $(\cvarfv,\cvarsv,\cvartv)$ in $\dlv^*$.

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A change of variables for an electrode tip domain. A change of variables $\cvarf$ maps a rectangular solid $\dlv^*$ (first panel) onto the electrode tip $\dlv$ (second panel). You can explore the mapping by moving with the mouse the red point in $\dlv^*$, which represents the point $(\cvarfv,\cvarsv,\cvartv)$, or the point blue point in $\dlv$, which represents the point $(x,y,z)$. When you move one point, the other point moves to reflect the mapping so that $(x,y,z)= \cvarf(\cvarfv,\cvarsv,\cvartv)$.

Since it is so late after you finished finding $\cvarf(\cvarfv,\cvarsv,\cvartv)$ and you
are tired, you almost start to integrate $g$ directly over the solid
$\dlv^*$. Fortunately, you realize that such a procedure doesn't make
sense because $g$ is a function of $(x,y,z)$ and hence is defined over
the electrode tip $\dlv$. Since it is not a function of $(\cvarfv,\cvarsv,\cvartv)$, you
can't integrate $g$ over the solid $\dlv^*$ in $(\cvarfv,\cvarsv,\cvartv)$-space. At this
point, you are too tired to figure out what you were supposed to do,
so you go to bed without finishing your calculation of what the
total charge on the electrode would be. You sleep fitfully, dreaming
of electrodes that were charged to dangerous levels attacking you.

The next morning, you wake up still groggy, but immediately realize
the solution to your problem. Although $g(x,y,z)$ is defined only on
your electrode $\dlv$, you can simply compose $g(x,y,z)$ with
$\cvarf(\cvarfv,\cvarsv,\cvartv)$ to obtain the function $g(\cvarf(\cvarfv,\cvarsv,\cvartv))$ that is
defined in terms of $(\cvarfv,\cvarsv,\cvartv)$. For a given point $(\cvarfv,\cvarsv,\cvartv)$ (shown by
the red point, above), the composition $g(\cvarf(\cvarfv,\cvarsv,\cvartv))$ gives the
density at the point $(x,y,z)=\cvarf(\cvarfv,\cvarsv,\cvartv)$ (shown by the blue
point, above) on the electrode $\dlv$. By integrating $g(\cvarf(\cvarfv,\cvarsv,\cvartv))$
over the solid $\dlv^*$ in $(\cvarfv,\cvarsv,\cvartv)$-space, you will integrate $g(x,y,z)$
over the solid $\dlv$ and compute the total charge. You feel pretty smug
that you had this realization even before breakfast and coffee.

Step 3: Include a factor to account for change in volume

Unfortunately, you are still too groggy after your fitful sleep to
remember everything you learned in multivariable calculus. You
completely forget that $(x,y,z)=\cvarf(\cvarfv,\cvarsv,\cvartv)$ will change volume in
$\dlv^*$ compared to volume in $\dlv$. Without compensating for this effect
of the map $\cvarf$, your calculations assume that volume in $\dlv$ is
the same as volume in $\dlv^*$.

Glancing at the above diagram is enough to guess what the result of
this error might be. The volume of $\dlv^*$ is larger than the volume of
$\dlv$. Remember that total charge equals charge density times volume.
If one uses the correct charge density $g$ of the electrode but multiplies
by volume in $\dlv^*$ rather than volume in $\dlv$, the calculation for the
total charge should be larger than the correct answer. If, for example,
$\dlv^*$ was uniformly twice as large as $\dlv$, then this incorrect
calculation for total charge would give an answer that was twice the
actual value.

Not realizing this mistake, you proceed with your incorrect
calculation and integrate $g(\cvarf(\cvarfv,\cvarsv,\cvartv))$ over $\dlv^*$ without
compensating for change in volume. Your result makes you so upset that
you kick the table in anger, hurting your foot and spilling your
coffee all over your calculations. But at that point, you don't
care. You rip up your coffee-soaked notes and limp back to bed
to cry. Your design is useless. The total charge on your electrode
is much too large. Your hard work is more likely to fry people's
brains than help them overcome epilepsy. Or so you think.

Once you stop crying, you discover that your throbbing foot has helped
lift your grogginess and your mind is actually becoming clear.
Finally, you remember the strict admonition of your calculus professor
to never forget to compensate for change in volume when changing
variables. You realize that maybe your electrode won't cook people's
brains and you might be able to salvage your design.

You grab a fresh piece of paper, race back to your table, and find a
dry spot on your table to work. You first sketch the following
diagram to show how dividing $\dlv^*$ into little boxes divides $\dlv$ into
small pieces. The diagram helps you visualize how changing variables changes
volume. You see that, especially around the critical tip of the
electrode, the volume in $\dlv$ is much smaller than the volume in $\dlv^*$.
This gives you hope that less charge may build up on the electrode
than you originally calculated.

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A change of variables for an electrode tip domain changes volume. A change of variables $\cvarf$ maps a rectangular solid $\dlv^*$ (first panel, in red) onto the the electrode tip $\dlv$ (second panel, in blue). It maps each small region of $\dlv^*$ onto a region of a different size in $\dlv$. You can move the highlighted red region in $\dlv^*$ and the highlighted blue region in $\dlv$ by dragging them with your mouse. As you move one region, the corresponding region in the other domain is highlighted. Notice that the red region has a different volume than the corresponding blue region. Especially when the blue region is near the electrode tip, the highlighted blue region is much smaller than the red region.

You realized that to calculate the integral
\begin{align*}
\iiint_\dlv g(x,y,z) \, dV,
\end{align*}
as an integral over $\dlv^*$ rather than $\dlv$, you need to make sure that
$dV$ still represents volume in $\dlv$ and not volume in $\dlv^*$. Instead
of replacing $dV$ with $d\cvarfv~d\cvarsv~d\cvartv$, as you originally did, you must
replace $dV$ with $| \det \jacm{\cvarf}(\cvarfv,\cvarsv,\cvartv)| ~d\cvarfv~d\cvarsv~d\cvartv$.
Just like for changing variables in double integrals, the absolute value of the
determinant of the partial derivative matrix of $\cvarf(\cvarfv,\cvarsv,\cvartv)$, sometimes called the volume expansion factor,
accounts for the stretching and shrinking of $\cvarf(\cvarfv,\cvarsv,\cvartv)$ as it
maps $\dlv^*$ onto $\dlv$.

You happily discover that the total charge on your electrode meets
acceptable limits and that your work is not destined for the trash
can! Overjoyed, you jump into the shower, don your best suit, and
rush to work. At the office, you slowly walk past the chief
scientist who asks if you determined the charge that will build up on
your electrode. You casually mention that the charge is well within
acceptable limits. You return to your desk to work on the next stage
of the design, vowing never to forget anything else you learned in
your multivariable calculus course.